Answer :
To analyze the given statements based on the equation [tex]\( a^{1 / n} = r \)[/tex], let's go through each one step-by-step:
### Statement A: [tex]\( \sqrt[n]{a} = r \)[/tex]
This statement asserts that the [tex]\( n \)[/tex]-th root of [tex]\( a \)[/tex] is equal to [tex]\( r \)[/tex].
- By definition, the [tex]\( n \)[/tex]-th root of [tex]\( a \)[/tex] can be written as [tex]\( \sqrt[n]{a} \)[/tex], which is equivalent to [tex]\( a^{1/n} \)[/tex].
- Since we are given that [tex]\( a^{1 / n} = r \)[/tex], this statement [tex]\( \sqrt[n]{a} = r \)[/tex] is indeed true.
### Statement B: [tex]\( r^n = a \)[/tex]
This statement claims that raising [tex]\( r \)[/tex] to the power of [tex]\( n \)[/tex] gives [tex]\( a \)[/tex].
- Given [tex]\( a^{1 / n} = r \)[/tex], we can raise both sides of this equation to the [tex]\( n \)[/tex]-th power to check the validity.
- [tex]\((a^{1/n})^n = r^n\)[/tex]
- Simplifying the left side, [tex]\( a = r^n \)[/tex].
Since [tex]\( r^n = a \)[/tex], this statement is true.
### Statement C: [tex]\( n^{1 / r} = a \)[/tex]
This statement asserts a relationship between [tex]\( n \)[/tex] and [tex]\( r \)[/tex] in which [tex]\( n \)[/tex] raised to the power of [tex]\( 1 / r \)[/tex] equals [tex]\( a \)[/tex].
- There is no direct mathematical property or transformation that relates [tex]\( a^{1/n} = r \)[/tex] to [tex]\( n^{1 / r} = a \)[/tex].
- The relationship given by [tex]\( a^{1 / n} = r \)[/tex] does not imply that [tex]\( n^{1 / r} = a \)[/tex].
Therefore, this statement is false.
### Statement D: [tex]\( d^{\prime} = n \)[/tex]
This statement introduces [tex]\( d' \)[/tex], which is not defined in the given context.
- Since [tex]\( d' \)[/tex] is not specified or related to the equation [tex]\( a^{1 / n} = r \)[/tex], we cannot establish any meaningful relationship between [tex]\( d' \)[/tex] and [tex]\( n \)[/tex].
Thus, this statement is also false.
### Conclusion
Based on the detailed analysis:
- Statement A is true.
- Statement B is true.
- Statement C is false.
- Statement D is false.
So, the correct responses are:
[tex]\[ \boxed{[1, 1, 0, 0]} \][/tex]
### Statement A: [tex]\( \sqrt[n]{a} = r \)[/tex]
This statement asserts that the [tex]\( n \)[/tex]-th root of [tex]\( a \)[/tex] is equal to [tex]\( r \)[/tex].
- By definition, the [tex]\( n \)[/tex]-th root of [tex]\( a \)[/tex] can be written as [tex]\( \sqrt[n]{a} \)[/tex], which is equivalent to [tex]\( a^{1/n} \)[/tex].
- Since we are given that [tex]\( a^{1 / n} = r \)[/tex], this statement [tex]\( \sqrt[n]{a} = r \)[/tex] is indeed true.
### Statement B: [tex]\( r^n = a \)[/tex]
This statement claims that raising [tex]\( r \)[/tex] to the power of [tex]\( n \)[/tex] gives [tex]\( a \)[/tex].
- Given [tex]\( a^{1 / n} = r \)[/tex], we can raise both sides of this equation to the [tex]\( n \)[/tex]-th power to check the validity.
- [tex]\((a^{1/n})^n = r^n\)[/tex]
- Simplifying the left side, [tex]\( a = r^n \)[/tex].
Since [tex]\( r^n = a \)[/tex], this statement is true.
### Statement C: [tex]\( n^{1 / r} = a \)[/tex]
This statement asserts a relationship between [tex]\( n \)[/tex] and [tex]\( r \)[/tex] in which [tex]\( n \)[/tex] raised to the power of [tex]\( 1 / r \)[/tex] equals [tex]\( a \)[/tex].
- There is no direct mathematical property or transformation that relates [tex]\( a^{1/n} = r \)[/tex] to [tex]\( n^{1 / r} = a \)[/tex].
- The relationship given by [tex]\( a^{1 / n} = r \)[/tex] does not imply that [tex]\( n^{1 / r} = a \)[/tex].
Therefore, this statement is false.
### Statement D: [tex]\( d^{\prime} = n \)[/tex]
This statement introduces [tex]\( d' \)[/tex], which is not defined in the given context.
- Since [tex]\( d' \)[/tex] is not specified or related to the equation [tex]\( a^{1 / n} = r \)[/tex], we cannot establish any meaningful relationship between [tex]\( d' \)[/tex] and [tex]\( n \)[/tex].
Thus, this statement is also false.
### Conclusion
Based on the detailed analysis:
- Statement A is true.
- Statement B is true.
- Statement C is false.
- Statement D is false.
So, the correct responses are:
[tex]\[ \boxed{[1, 1, 0, 0]} \][/tex]
